Tiers are the old badges : Challenger, Diamond...... Bronze. There are 5 divisions in each tier (from I being the best to V being the worst) except for challenger.

You can go up a division by getting 100 LP and winning a bo3.

You can go up a tier by getting 100 LP on Division 1 of your tier, then winning a bo5.

You can go down a division by being at 0 LP in your division (between I and IV) and losing several games.

You can't go down a tier (from gold to silver for instance), even if you're the lowest player in Division V and lose several games.
EDIT : you can actually go down a tier in the event you do not play for 28 days straight.

YOU CAN DROP A TIER. It happened to me. I was in SIlver, I took a break for the game, so Riot punished me. I came back and am now in Bronze 1. I found it ridiculous, you should drop tiers due to losing, not due to NOT playing. Sure, take my LP, drop me to the bottom of my tier, but don't make me drop actual tiers just because I took a break.

If you want to read about it, Riot addresses this in the "Competitive" - "Season 3 FAQ" - "What if I don't play" or something...anyways, here is a direct quote:

After 28 days of inactivity in a particular ranked queue, a couple things will happen. First, you’ll be hidden from the standings in your League, and you may lose League Points depending on your current tier. Every seven days thereafter, you’ll lose League Points again until you play a match in that queue.

The number of League Points lost following each period of decay is as follows:

Diamond: 50
Platinum: 35
Gold: 25
Silver: 10
Bronze: 0

If you fall below zero League Points as a result of decay, you’ll be placed into the next lowest division. If you’re already in Division V of your league, you’ll fall into a new league in the next tier down. This is currently the only way to get demoted to a lower tier.